Practical compressive sensing with Toeplitz and circulant matrices

Compressive sensing encodes a signal into a relatively small number of incoherent linear measurements. In theory, the optimal incoherence is achieved by completely random measurement matrices. However, such matrices are often difficult and costly to implement in hardware realizations. Random Toeplitz and circulant matrices can be easily (or even naturally) realized in various applications. This paper introduces fast algorithms for reconstructing signals from incomplete Toeplitz and circulant measurements. Computational results are presented to show that Toeplitz and circulant matrices are not only as effective as random matrices for signal encoding, but also permit much faster decoding.